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Limiting distributions of theta series on Siegel half-spaces


Authors: F. Götze and M. Gordin
Translated by:
Original publication: Algebra i Analiz, tom 15 (2003), nomer 1.
Journal: St. Petersburg Math. J. 15 (2004), 81-102
MSC (2000): Primary 11Fxx, 37D30
DOI: https://doi.org/10.1090/S1061-0022-03-00803-3
Published electronically: December 31, 2003
MathSciNet review: 1979719
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $m \ge 1$ be an integer. For any $Z$ in the Siegel upper half-space we consider the multivariate theta series

\begin{displaymath}\Theta (Z)= \sum _{{\overline{n}} \in \mathbb{Z}^{m}} \exp (\pi i \,{}^{t} {\overline{n}} Z {\overline{n}}).\end{displaymath}

The function $\Theta $ is invariant with respect to every substitution $Z\longmapsto Z + P$, where $P$ is a real symmetric matrix with integral entries and even diagonal. Therefore, for any real matrix $Y > 0$ the function $\Theta _{Y} ( \cdot ) = (\det Y)^{1/4} \Theta (\cdot +iY)$ can be viewed as a complex-valued random variable on the torus $\mathbb{T}^{m(m+1)/2}$ with the Haar probability measure. It is proved that the weak limit of the distribution of $ \Theta _{\tau Y}$ as $\tau \to 0$ exists and does not depend on the choice of $Y$. This theorem is an extension of known results for $m=1$ to higher dimension. Also, the rotational invariance of the limiting distribution is established. The proof of the main theorem makes use of Dani-Margulis' and Ratner's results on dynamics of unipotent flows.

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Additional Information

F. Götze
Affiliation: Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Email: goetze@mathematik.uni-bielefeld.de

M. Gordin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191011, Russia
Email: gordin@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-03-00803-3
Keywords: Theta series, Siegel's half-space, convergence in distribution, closed horospheres, unipotent flows
Received by editor(s): September 2, 2002
Published electronically: December 31, 2003
Additional Notes: Supported in part by the DFG-Forschergruppe FOR 399/1-1.
M. Gordin was also partially supported by RFBR (grant no. 02.01-00265) and by Sc. Schools grant no. 2258.2003.1. He was a guest of SFB-343 and the Department of Mathematics at the University of Bielefeld while the major part of this paper was prepared.
Article copyright: © Copyright 2003 American Mathematical Society

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